---
tag: 0ZB7J071WQ
kind: statement
role: proposition
name: A complex number times its conjugate
msc: [30A99]
uses: [0BJV24C4SY, 0KW2X0TFBX]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_conjugate
    locator: Properties
---

## Statement

Let $z$ be a complex number, let $\bar{z}$ be its [[conjugate|0BJV24C4SY]] and let $|z|$ be its [[modulus|0KW2X0TFBX]].
Then
$$z \bar{z} = |z|^2.$$
In particular, $z \bar{z}$ is a nonnegative real number.

## Remarks

With $z = a + b i$: $(a + b i)(a - b i) = a^2 - (b i)^2 = a^2 + b^2$.
This identity gives the formula for the [[inverse of a complex number|0FQRAV5SKH]].
It also gives a quick proof that the [[modulus is multiplicative|0HYNZE940T]].

## Explanations

### L1 human

When you multiply a complex number by its conjugate, the imaginary parts cancel.
What remains is the square of the length of the number, for example $(3 + 4 i)(3 - 4 i) = 9 + 16 = 25$.